CR · AC power

RMS Value and Average Power

An AC voltage rises and falls without rest, so it is hard to name it with a single value. Yet mains voltage in much of the world is quoted as 220V; this unit explains what that number means and why it lets power be calculated as if the voltage were DC.

Find the DC that heats the same

The curve is a sinusoidal voltage, and the horizontal line drawn across it is the RMS value. Raise the amplitude: the peaks climb all the way to the amplitude, yet in terms of heating a resistor the wave equals a single DC value smaller than the amplitude. Set the amplitude so that this RMS value reaches the target of 220V.

Amplitude V_peakV_peak = 140 V

RMS and peak

V_rms = 99 V  (= V_peak/√2)

Far from 220

Instantaneous power keeps sloshing

The instantaneous power delivered to a resistor is the voltage at that instant squared and divided by the resistance, v²/R. When the voltage passes through 0 the power is 0 too, and at the peaks the power is greatest. Because of the square, the power is always positive even when the voltage is negative. To describe this fluctuating power with one value, you have to average it over a full cycle.

The mean of the square is one half

For a sine wave v = V_peak sin, the square swings between the peak V_peak² and 0, and its average over one cycle is exactly half of V_peak². The average power is therefore (V_peak²/2)/R. To write this like DC as P = V_rms²/R, we define V_rms² = V_peak²/2, which means V_rms = V_peak/√2.

The RMS value is the equivalent DC

The real meaning of the RMS value is the DC voltage that delivers the same average power to the same resistor. A 220V AC supply heats a lamp exactly as much as 220V DC would. So when mains voltage is called 220V, it refers not to the peak (about 311V) but to this RMS value. Converting to RMS makes AC power calculations as simple as DC ones.

ObserveV_rms = V_peak / √2

The RMS value is the amplitude divided by √2.

ChooseP = ?

With RMS values, calculate just as for DC.

Fill in<v²> = V_peak² / ?

The average of the square is half.

On your ownV_rms = ? · V_peak

1/√2 is about 0.707.

Back to the first screen

As you raised the amplitude, the RMS line always followed √2 times below the peak, and the RMS value reached 220V when the peak was about 311V. Though the waveform swings without rest, in its ability to heat a resistor it was exactly the same as a single DC value of 220V. Translating the swinging into DC with a single number, the RMS value, makes AC power easy to handle.

The RMS value converts a sinusoid into the DC value that delivers the same average power: V_rms = V_peak/√2 ≈ 0.707 V_peak. The average power is the instantaneous power v²/R averaged over one cycle, and since the average of the square is V_peak²/2, it is calculated like DC as P = V_rms²/R. A 220V mains rating refers to this RMS value, not to the peak.